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Those Fascinating Numbers
Page113(133 of 451)

Those Fascinating Numbers 113 936 • the second number n such that (σ(n)+ γ(n))/n is an integer: the only numbers n 109 satisfying this property are 6, 936 and 1 638. 937 • the third prime number q such that ∑ p≤q p is a multiple of 100: here this sum is equal to 67400 (see the number 563). 942 • the second solution of σ2(n) = σ2(n + 10) (see the number 120). 944 • the smallest number n such that n and n + 1 each have five prime factors counting their multiplicity: 944 = 24 · 59 and 945 = 33 · 5 · 7 (see the number 135). 945 • the smallest odd abundant number: we have σ(945) = σ(33 · 5 · 7) = 945 + 975; the odd abundant numbers119 104 are 945, 1575, 2205, 2835, 3465, 4095, 4725, 5355, 5775, 5985, 6435, 6615, 6825, 7245, 7425, 7875, 8085, 8415, 8505, 8925, 9135, 9555 and 9765 (see also the number 5 391 411 025); • the smallest of the eight existing primitive non deficient numbers: we say that a number n is non deficient if σ(n) ≥ 2n and we say that it is primitive non deficient if it is non deficient and if it is not a multiple of a smaller non deficient number (see Dickson [67], p. 31): these eight numbers are 945, 1 575, 2 205, 7 425, 78 975, 131 625, 342 225 and 570 375. 946 • the fourth number n such that 2n ≡ −2 (mod n) (Schinzel, see R.K. Guy [101], F10): the numbers n 106 satisfying this property are 2, 6, 66, 946, 8 646, 180 246, 199 606, 265 826 and 383 846. 952 • the fourth number that is equal to the sum of the third power of its digits added to the product of its digits: the only numbers satisfying this property120 are 31, 370, 407 and 952. 119Observe that the first eight terms ak of this sequence are given by ak = 945 + 630k, k = 0, 1, 2, . . . , 7. 120One can argue, as in the footnote tied to the number 1 324, that any such number can have at most five digits; using a computer, it is then easy to prove that 952 is the largest number with this property.